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Properties

Label 2-252-63.41-c3-0-9
Degree $2$
Conductor $252$
Sign $0.976 + 0.213i$
Analytic cond. $14.8684$
Root an. cond. $3.85596$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.37 − 2.80i)3-s + (−2.20 + 3.82i)5-s + (13.5 − 12.6i)7-s + (11.3 + 24.5i)9-s + (−59.1 + 34.1i)11-s + (29.9 + 17.3i)13-s + (20.3 − 10.5i)15-s + 21.8·17-s − 124. i·19-s + (−94.6 + 17.2i)21-s + (61.3 + 35.4i)23-s + (52.7 + 91.3i)25-s + (19.2 − 138. i)27-s + (187. − 108. i)29-s + (242. + 140. i)31-s + ⋯
L(s)  = 1  + (−0.842 − 0.539i)3-s + (−0.197 + 0.341i)5-s + (0.732 − 0.681i)7-s + (0.418 + 0.908i)9-s + (−1.62 + 0.936i)11-s + (0.639 + 0.369i)13-s + (0.350 − 0.181i)15-s + 0.311·17-s − 1.50i·19-s + (−0.983 + 0.178i)21-s + (0.556 + 0.321i)23-s + (0.422 + 0.731i)25-s + (0.137 − 0.990i)27-s + (1.20 − 0.694i)29-s + (1.40 + 0.811i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.213i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.976 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.976 + 0.213i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.976 + 0.213i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.297907981\)
\(L(\frac12)\) \(\approx\) \(1.297907981\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (4.37 + 2.80i)T \)
7 \( 1 + (-13.5 + 12.6i)T \)
good5 \( 1 + (2.20 - 3.82i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (59.1 - 34.1i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-29.9 - 17.3i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 21.8T + 4.91e3T^{2} \)
19 \( 1 + 124. iT - 6.85e3T^{2} \)
23 \( 1 + (-61.3 - 35.4i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-187. + 108. i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-242. - 140. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 150.T + 5.06e4T^{2} \)
41 \( 1 + (-136. + 236. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (136. + 236. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-97.3 - 168. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 520. iT - 1.48e5T^{2} \)
59 \( 1 + (-301. + 521. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-145. + 84.0i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-371. + 643. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 758. iT - 3.57e5T^{2} \)
73 \( 1 - 1.15e3iT - 3.89e5T^{2} \)
79 \( 1 + (-78.6 - 136. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (137. + 237. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.04e3T + 7.04e5T^{2} \)
97 \( 1 + (-211. + 122. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.37662696070467793293157010096, −10.83003292519615522916886202363, −10.01133618158282780560887891933, −8.347319652543847224172785882964, −7.40718459656956004303151004782, −6.78031767117745008370291961823, −5.26812023652145089003674782524, −4.52558055700170806448814239877, −2.53465011960203998706779191321, −0.907272773553533008949393795954, 0.856714830289940194090642475134, 2.97259380349265123332893547061, 4.54934903768388780186816796889, 5.45069178860192775599107940231, 6.20818300898302383236417495626, 8.040834573080652638918437767844, 8.488053438539324315024762679488, 10.01815474087429858234336914161, 10.70050757404843814867093232912, 11.57043087334080292654826949890

Graph of the $Z$-function along the critical line